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María Chara

Publications and source records attributed to María Chara.

8 recordsLinked to original sources
arXiv2026

Lifting iso-dual algebraic geometry codes

In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field $\mathbb{F}_q$ with $q$ elements. Given a finite separable extension $\mathcal{M}/\mathcal{F}$ of function fields and an iso-dual AG-code $\mathcal{C}$ defined over $\mathcal{F}$, we provide a general method to lift the code $\mathcal{C}$ to another iso-dual AG-code $\tilde{\mathcal{C}}$ defined over $\mathcal{M}$ under some assumptions on the divisors $D$ and $G$ and on the parity of the involved different exponents. We apply this method to lift iso-dual AG-codes over the rational function field to elementary abelian $p$-extensions, like the maximal function fields defined by the Hermitian, Suzuki, and one covered by the $GGS$ function field. We also obtain long binary and ternary iso-dual AG-codes defined over cyclotomic extensions.

cs.IT↗
arXiv2025

Bounds for the relative class number problem for function fields

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the relative class number two problem in all cases where the base field has constant field not equal to $\mathbb{F}_2$.

math.NT↗
arXiv2025

Good iso-dual AG-codes from towers of function fields

We present a simple method to establish the existence of asymptotically good sequences of iso-dual AG-codes. A key advantage of our approach, beyond its simplicity, is its flexibility, allowing it to be applied to a wide range of towers of function fields. As a result, we present a novel example of an asymptotically good sequence of iso-dual AG-codes over a finite field with 8 elements.

math.NT↗
arXiv2022

Minimum Distance and Parameter Ranges of Locally Recoverable Codes with Availability from Fiber Products of Curves

We construct families of locally recoverable codes with availability $t\geq 2$ using fiber products of curves, determine the exact minimum distance of many families, and prove a general theorem for minimum distance of such codes. The paper concludes with an exploration of parameters of codes from these families and the fiber product construction more generally. We show that fiber product codes can achieve arbitrarily large rate and arbitrarily small relative defect, and compare to known bounds and important constructions from the literature.

cs.IT↗
arXiv2021

The conorm code of an AG-code

Given a suitable extension $F'/F$ of algebraic function fields over a finite field $\mathbb{F}_q$, we introduce the conorm code $\text{Con}_{F'/F}(\mathcal{C})$ defined over $F'$ which is constructed from an algebraic geometry code $\mathcal{C}$ defined over $F$. We study the parameters of $\text{Con}_{F'/F}(\mathcal{C})$ in terms of the parameters of $\mathcal{C}$, the ramification behavior of the places used to define $\mathcal{C}$ and the genus of $F$. In the case of unramified extensions of function fields we prove that $\text{Con}_{F'/F}(\mathcal{C})^\perp = \text{Con}_{F'/F}({\mathcal{C}}^\perp)$ when the degree of the extension is coprime to the characteristic of $\mathbb{F}_q$. We also study the conorm of cyclic algebraic-geometry codes and we show that some repetition codes, Hermitian codes and all Reed-Solomon codes can be represented as conorm codes.

math.NT↗
arXiv2021

On cyclic algebraic-geometry codes

In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms. We prove that cyclic algebraic geometry codes constructed in this way are closely related to cyclic extensions. We also give a detailed study of the monomial equivalence of cyclic algebraic geometry codes constructed with our method in the case of a rational function field.

math.AG↗
arXiv2019

Block-transitive algebraic geometry codes attaining the Tsfasman-Vladut-Zink bound

We study the asymptotic behavior of a family of algebraic geometry codes, which we call block-transitive, that generalizes the classes of transitive and quasi-transitive codes. We prove, by using towers of algebraic function fields, that there are sequences of codes in this family attaining the Tsfasman-Vladut-Zink bound over finite fields of square cardinality. We give the exact length of these codes as well as explicit lower bounds for their parameters.

math.NT↗